On a New Class of Univariate Continuous Distributions that are Closed Under Inversion

Inverted probability distributions find applications in various real – life situations including econometrics, survey sampling, biological sciences and life – testing. Closure under inversion implies that the reciprocal of a continuous random variable X has the same probability function as the original random variable, allowing for a possible change in parameter values. To date, only a very few probability distributions have been found to possess the closure property. 

In this paper, an attempt has been made to generate a class of distributions that are closed under inversion, and to develop some statistical properties of this class of distributions.


Introduction
Inverted probability distributions find applications in various real -life situations including econometrics, survey sampling, biological sciences, engineering sciences, and , most prominently, in life -testing.As such, various authors have derived a variety of inverted distributions, and have developed their statistical properties.
Closure under inversion implies that the reciprocal of a continuous random variable X has the same probability function as the original random variable, allowing for a possible change in parameter values.In case the parameter values are identical to those of the original distribution, the random variable X (and its reciprocal) will be said to be Strictly Closed Under Inversion.
To date, only a very few probability distributions have been found to possess the closure property.For example, the Cauchy (0, 1) distribution is closed under inversion in the strict sense, whereas the ( ) distribution is closed in the generalized sense.
In this paper, an attempt has been made to generate a class of distributions that are closed under inversion, and to establish some of the fundamental properties of this particular class of distributions.

Definition
A probability density function f(x) will be said to be closed under inversion if the form of the probability density function of 1 / X is the same as that of f(x).In case the parameters of the inverted distribution are identical to those of the original distribution, the random variable X (and its reciprocal) will be said to be strictly closed under inversion.

A Class of Distributions that are Strictly Closed Under Inversion
With reference to the development of a class of distributions that are Strictly Closed Under Inversion, we present the following theorem: As far as the point regarding f(x) being a proper pdf is concerned, any function of the form given by (3.1) will be a proper pdf as long as f(x) > 0 over its domain, and the integral of the function over its domain is convergent.

Alternative Proof
If the function f(x) is Strictly Closed Under Inversion, then it satisfies the functional equation No ow w e eq q ( (3 3. .1 1) ) c ca an n b be e w wr ri it tt te en n a as s Replacing x by 1/x, we obtain Hence the function f(x) given by (3.1) is Strictly Closed Under Inversion.
2. If we let a -> 1, then 1/a -> 1 and f(x) is degenerate.(In other words, if a -> 1, f(x) is a one -point distribution located on x = 1.)

Examples
The class of SCUI distributions given by (3.1) encompasses a variety of probability density functions, some of which are presented in Table 1:  ( ) The well -known F distribution with v1 = v2 = v i.e.

Corollary No. 1
For every pdf belonging to the class of SCUI distributions given by eq.(3.1) above, the median is equal to unity i.e. ~ X = X 0.5 = 1

Remark
The converse of this result is not generally true.There do exist continuous distributions that are defined on (0, infinity) and the median of which is unity but which are not Strictly Closed Under Inversion.

Corollary No. 2
For every pdf belonging to the class of SCUI distributions given by eq.(3.1) above, the area under the curve between X q and 1/X q is equal to 1 -2q i.e.P[ X q < X < 1/X q ] = 1-2q Proof: The proof is simple.

Remark
This result is somewhat comparable with the well -known result that, for a normal distribution with mean Mu and standard deviation Sigma: The class of distributions of non -negative random variables given in this paper is not exhaustive.There exists at least one probability density function that extends from -infinity to infinity and is closed under inversion i.e. the Cauchy (0, 1) distribution.
All of the above is true under Regularity Conditions such as absolute continuity, absolute differentiability, etc.
. and Sheikh, A.K. (1981).Reliability Computation for Bernstein Distribution Strength and Stress.Submitted to the 10 th Pak.Statistical Conference held in Islamabad, Pakistan.